Quantum wells, superlattices, and other quantum heterostructures comprised of thin semiconductor layers deposited by molecular beam epitaxy (MBE), metalorganic chemical vapor deposition (MOCVD), or some other process, are important components in a number of optoelectronic devices. Many of the unique and advantageous properties of semiconductor quantum heterostructures result from the ability to tune (or engineer) their electronic band structure, and other properties that depend on the electronic band structure, by varying the layer thicknesses and elemental, binary, or alloy layer compositions. Of the optoelectronic properties that can be varied, some of the most important for many applications such as lasers, light detection, solar cells for power generation, or thermophotovoltaics, are the optical absorption coefficient, the spontaneous emission rate, and the carrier concentration as a function of the quasi-Fermi levels. Accurate modeling of these quantities in layered materials therefore represents an important component of device simulations that are needed for materials/device design, device optimization, and data interpretation.
Superlattices are typically classified as being either type-I or type-II. In a type-I structure (such as GaAs/AlGaAs), both the conduction band minimum and valence band maximum reside in the same material (GaAs in this example). Therefore, the electron and hole wavefunctions are concentrated mostly in the same material (GaAs), hence they tend to overlap strongly for a large optical matrix element. On the other hand, a type-II superlattice (such as InAs/GaInSb) has its conduction band minimum in one material (InAs) while the valence band maximum is in another (GaInSb). In that case, the electron and hole wavefunctions do not necessarily overlap strongly and the optical matrix element may be much smaller. Type-I and type-II superlattices may also have different symmetries, with the two-layer type-II superlattice and the three-layer type-II quantum well being classic examples of asymmetric systems.
To illustrate the implementation and advantages of the invention, the following description treats the particular example of a type-II antimonide superlattice. However, the invention is similarly advantageous when applied to most type-II superlattices, and some features substantially reduce the run times required to compute the absorption coefficients, radiative emission rates, carrier densities, and other properties for type-I quantum wells and superlattices as well.
Antimonide type-II superlattices are promising as the absorber material for long-wave infrared (LWIR), very long-wave infrared (VLWIR), and mid-wave infrared (MWIR) photodetectors. These III-V-based photodetectors have a number of potential advantages over the incumbent InSb and HgCdTe technologies, such as higher-temperature operation, improved wafer-scale manufacturability, and larger substrates. However, the understanding of the type-II superlattices, which can be comprised of two-constituent InAs/Ga(In)Sb and InAs/InAsSb designs as well as somewhat more complicated multi-constituent structures, remains incomplete at this date. In order to realize the full potential of the type-II superlattices, their absorption characteristics must be carefully modeled, since these play the dominant role in determining the quantum efficiency (QE) of the photodetector when the diffusion length is comparable to or greater than the absorption depth. Furthermore, in many circumstances it is important to model the radiative lifetime in these superlattices in order to estimate its contribution to the total lifetime, which in turn determines the dark current density of the photodetectors. The radiative lifetime is also important for estimating the efficiency of light-emitting diodes (LEDs), the threshold for lasers, and other properties of optoelectronic devices.
In order to evaluate the dark current densities in photodiode and photovoltaic structures, it is also important to calculate the intrinsic carrier density as well as the variation of the quasi-Fermi level position with carrier density. The latter is needed when the doping is so high that the carrier density is degenerate, or in some cases, to calculate the contribution of a particular trap with an energy inside the band gap to the Shockley-Read recombination rate. While reasonable guesses can be obtained using the parabolic approximation with the appropriate density-of-states effective mass, a more accurate evaluation is needed in many circumstances.
The absorption coefficient α(ω), reflecting the percentage of the incident light that is absorbed, is an important characteristic of a semiconductor superlattice used for such photodiode and photovoltaic structures and can be calculated from the following equation:
                              α          ⁡                      (                                          h                _                            ⁢                                                          ⁢              ω                        )                          =                                            4              ⁢                                                          ⁢                              π                2                            ⁢                              α                0                                                    n              r                                ⁢                      1                                          h                _                            ⁢                                                          ⁢              ω                                ⁢                                    ∑                              n                ,                m                                      ⁢                                          ∫                                  -                                      π                    d                                                                    +                                      π                    d                                                              ⁢                                                                    ⅆ                                          k                      z                                                                            2                    ⁢                                                                                  ⁢                    π                                                  ⁢                                                      ∫                    0                                          2                      ⁢                                                                                          ⁢                      π                                                        ⁢                                                            ⅆ                      φ                                        ⁢                                                                  ∫                        0                        ∞                                            ⁢                                                                                                                                  k                                                                                      ⁢                                                          ⅆ                                                              k                                                                                                                                                                          4                            ⁢                                                                                                                  ⁢                                                          π                              2                                                                                                      ⁢                                                                                                  ⁢                                                                                                                                                                          M                                ⁡                                                                  (                                  k                                  )                                                                                                                                                    2                                                    [                                                                                                          ⁢                                                                                                                    f                                m                                                            ⁡                                                              (                                                                                                      E                                    m                                                                    ⁡                                                                      (                                    k                                    )                                                                                                  )                                                                                      -                                                                                                     ⁢                                                                              f                            n                                                    (                                                                                                          ⁢                                                                                    E                              n                                                        (                                                                                                                  ⁢                            k                            )                                                    )                                                                                                                                                            ]        ⁢                  ⁢          L      (                          ⁢                                    E            n                    (                                          ⁢          k          )                -                              E            m                    ⁡                      (            k            )                          -                              h            _                    ⁢                                          ⁢          ω                    )        ,
where ω is the photon energy of the light incident on the superlattice; α0 is the fine-structure constant; nr is the refractive index of the absorbing material; |M|2 is the square of the optical matrix element given by |∇kH|2, where H is the band structure Hamiltonian; m is the valence subband and n is the conduction subband; L is the lineshape function (see S. W. Corzine, R.-H. Yang, and L. A. Coldren, “Optical Gain in III-V Bulk and Quantum-Well Semiconductors”, Chap. 1 in Quantum Well Lasers, ed. by P. S. Zory, Academic, New York 1993), which depends on the difference between the electron energy En and the hole energy Em at wavevector k; fm and fn are hole and electron Fermi functions, respectively; and d is the superlattice period.
Generally, the absorption coefficient derived from Equation (1) is relatively insensitive to the precise form assumed for the lineshape function, which may be approximated as a delta function, Lorentzian, Gaussian, 1/cos h(x) function, or some other suitable form. If population inversion is established, i.e., fm−fn<0, the absorption coefficient takes on a negative value and in such a case is commonly referred to as the “gain coefficient.” This coefficient is important in simulations of laser properties where the gain must be calculated.
While in principle it is straightforward to derive the absorption coefficient from the superlattice band structure and optical matrix elements (see J. R. Meyer, C. A. Hoffman, F. J. Bartoli, and L. R. Ram-Mohan, “Type-II quantum-well lasers for mid-wavelength infrared,” Appl. Phys. Lett. 67, 757 (1995)), the existing methods for doing so require integration over the band structure in a three-dimensional region of k space. That is, three integrations must be carried out in k space,
      i    .    e    .    ,            ∫              -                  π          d                            π        d              ⁢                  ⅆ                  k          z                            2        ⁢                                  ⁢        π              ,            ∫      0              2        ⁢                                  ⁢        π              ⁢          ⅆ      φ        ,      and    ⁢                  ⁢                  ∫        0        ∞            ⁢                                                  k                                      ⁢                          ⅆ                              k                                                                          4            ⁢                          π              2                                      .            This requires accumulating a large number of band structure data points covering the entire region so that the integration procedure converges. The band structure data points for such quantities as Ei(k) and M(k) are obtained by diagonalizing the Hamiltonian matrix and post-processing the resulting eigenvalues and eigenvectors at each value of k.
Any superlattice having barriers sufficiently thin that the structure is not effectively a quantum well can exhibit significant dispersion along the vertical (z) direction. Furthermore, because the dispersions of the various electron and hole subbands may vary significantly with in-plane direction, using conventional procedures one must evaluate the various quantities such as the energy for each subband Ei(k) and the optical matrix element M(k) at a large number of points in k space as the three integrations are performed numerically. Because these evaluations, which require diagonalizing the Hamiltonian using the 8-band k·p or some other method, can consume a great deal of computer run time, the full numerical evaluation of all three integrals can be extremely slow.